Non-Unique Factorizations

Non-Unique Factorizations : Algebraic, Combinatorial and Analytic Theory

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From its origins in algebraic number theory, the theory of non-unique factorizations has emerged as an independent branch of algebra and number theory. Focused efforts over the past few decades have wrought a great number and variety of results. However, these remain dispersed throughout the vast literature. For the first time, Non-Unique Factorizations: Algebraic, Combinatorial, and Analytic Theory offers a look at the present state of the theory in a single, unified resource. Taking a broad look at the algebraic, combinatorial, and analytic fundamentals, this book derives factorization results and applies them in concrete arithmetical situations using appropriate transfer principles. It begins with a basic introduction that can be understood with knowledge of standard basic algebra. The authors then move to the algebraic theory of monoids, arithmetic theory of monoids, the structure of sets of lengths, additive group theory, arithmetical invariants, and the arithmetic of Krull monoids. They also provide a self-contained introduction to abstract analytic number theory as well as a modern treatment of W. Narkiewicz's analytic theory of non-unique factorizations. Non-Unique Factorizations: Algebraic, Combinatorial, and Analytic Theory builds the discussion from first principles to applied problem solving, making it ideally suited to those not familiar with the theory as well as those who wish to deepen their understanding.show more

Product details

  • Hardback | 728 pages
  • 177.8 x 251.5 x 43.2mm | 1,360.79g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 7 black & white illustrations
  • 1584885769
  • 9781584885764

Table of contents

CONCEPTS IN FACTORIZATION THEORY AND EXAMPLES Atoms and Primes Free Monoids, Factorial Monoids and Factorizations BF-Monoids Systems of Sets of Lengths FF-Monoids The Catenary Degree and the Tame Degree Rings of Integers of Algebraic Number Fields ALGEBRAIC THEORY OF MONOIDS v-Ideals Prime Ideals and Localizations Complete Integral Closures and Krull Monoids Divisor Homomorphisms and Divisor Theories Krull Monoids and Class Groups Defining Systems and v-Noetherian Monoids Finitary Monoids Class Semigroups C-Monoids and Finitely Primary Monoids Integral Domains Congruence Monoids and Orders ARITHMETIC THEORY OF MONOIDS Finitary Monoids Transfer Principles C-Monoids Saturated Submonoids and Krull Monoids Type Monoids Faithfully Saturated Submonoids Integral Domains and Congruence Monoids Factorizations of Powers of an Element THE STRUCTURE OF SETS OF LENGTHS Multidimensional Arithmetical Progressions Almost Arithmetical Multiprogressions An Abstract Structure Theorem for Sets of Lengths Pattern Ideals and Complete s-Ideals in Finitary Monoids Products of Strongly Primary Monoids and their Submonoids C-Monoids Integral Domains and Congruence Monoids Realization Theorems and Further Examples Sets of Lengths of Powers of an Element ADDITIVE GROUP THEORY Sequences over Abelian Groups Addition Theorems Zero-Sumfree Sequences Cyclic Groups Group Algebras and p-Groups Coverings by Cosets and Elementary p-Groups Short Zero-Sum Sequences and the Inductive Method Groups of Rank Two ARITHMETICAL INVARIANTS OF KRULL MONOIDS The Generalized Davenport Constants The Narkiewicz Constants The Elasticity and Its Refinements The Catenary Degree The Tame Degree Sets of Lengths Containing 2 The Set of Distances and Maximal Half-Factorial Sets Minimal Non-Half-Factorial Sets GLOBAL ARITHMETIC OF KRULL MONOIDS Arithmetical Characterizations of Class Groups I Arithmetical Characterizations of Class Groups II The System of Sets of Lengths for Finite Abelian Groups The System of Sets of Lengths for Infinite Abelian Groups Additively Closed Sequences and Restricted Sumsets Factorization of Large Elements ABSTRACT ANALYTIC NUMBER THEORY Dirichlet Series A General Tauberian Theorem Abstract Formations and Zeta Functions Arithmetical Formations I: Zeta Functions Arithmetical Formations II: Asymptotic Results Arithmetical Formations III: Structure Theory Geometrical Formations I: Asymptotic Results Geometrical Formations II: Structure Theory Algebraic Function Fields Obstructed Formations ANALYTIC THEORY OF NON-UNIQUE FACTORIZATIONS Analytic Theory of Types Elements with Prescribed Factorization Properties The Number of Distinct Factorizations Block-Dependent Factorization Properties APPENDIX A: ABELIAN GROUPS APPENDIX B: COMPLEX ANALYSIS APPENDIX C: THEORY OF INTEGRATION APPENDIX D: POLYHEDRAL CONES BIBLIOGRAPHY LIST OF SYMBOLS SUBJECT INDEXshow more

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